On group vertex magic graphs

Let G=(V(G),E(G)) be a simple undirected graph and let A be an additive abelian group with identity 0. A mapping l:V(G)-> A\{0} is said to be a A-vertex magic labeling of G if there exists an element mu of A such that w(v)=Sigma u is an element of N(v)l(u)=mu for any vertex v of G, where N(v) is the open neighborhood of v. A graph G that admits such a labeling is called an A-vertex magic graph. If G is A-vertex magic graph for any nontrivial abelian group A, then G is called a group vertex magic graph. In this paper, we obtain a few necessary conditions for a graph to be group vertex magic. Further, when AZ2 circle plus Z2, we give a characterization of trees with diameter at most 4 which are A-vertex magic.